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A new closed-loop subspace identification
Santos D. Miranda Borjas, Elmer Rolando Llanos Villareal Depto de Ciências Exatas e Naturais, DCEN, UFERSA
59625900, Av. Francisco Mota, 572 Barrio Costa e Silva, Mossoró, RN E-mail: [email protected]
E-mail: [email protected]
Claudio Garcia,
Depto de Telecomunicação e Controle,PTC, USP 05508-970, Av. Avenida Prof. Luciano Gualberto, travessa 3 nº 380, Barrio Butantã, São Paulo, SP
E-mail: [email protected]
Abstract: There are different methods to closed-loop subspace identification. For example, the MOESP and N4SID methods. Based on them, it is introduced the MON4SID method, which estimates the extended observability matrix and the state sequence directly from a LQ decomposition. This new method uses an algorithm to identify state space model of a plant in a closed-loop system, in the same way as in MOESP method. The advantage of the proposed algorithm is that it does not require any knowledge about the controller, whereas such information is essential for other methods (e.g. N4SID). Keywords: Closed-loop identification; state space models; subspace identification.
1. INTRODUCTION
Among the subspace methods it is worth to mention the Multivariable Output-Error State sPace (MOESP) [18], Canonical Variate Analysis (CVA) [9] and Numerical algorithms for Subspace State Space System Identification (N4SID) [15]. Given the input and output data, these methods can provide satisfactory estimates of models operating in open-loop systems. One of the main assumptions of theses methods is that the process and the measurement noises are independent of the plant input. This assumption is violated when the system is working in closed-loop. The closed-loop identification is of special interest for a large number of engineering applications [2], [10], [12]. Several closed-loop identification methods have been suggested in the last years and can be broadly classified into three main groups: direct, indirect and joint input / output identification methods [1]. The results of any of the aforesaid N4SID, MOESP and CVA methods are asymptotically biased when closed-loop identification is applied. To solve this problem, the MOESP method [17] proposed a closed-loop subspace identification method through the identification of an overall open-loop state space. Based on it, the plant and the controller models are estimated. To do so, it is necessary to know the order of the controller. In the N4SID case [16], it is necessary to know a limited number of impulse response samples of the controller and, via direct identification, the plant model is estimated. There are other possible solutions for the closed-loop identification problem; the reader can consult [4], [7], [8], [11], [13].
Combining the MOESP and N4SID methods, we obtain the MON4SID algorithm, which estimates the extended observability matrix in the same way it occurs in the MOESP method, the state sequence is computed through the oblique projection as it is done in the N4SID method. From this sequence, the past and future states are obtained and finally a consistent estimate of the matrices system is obtained, applying the least squares method. In this paper, it is proposed an algorithm to identify the state space model of a plant in a closed-loop system, in the same way as it was proposed in the MOESP method,
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that first computes a global model from which is extracted the plant model. This method does not need any knowledge about the controller.
1.1 Open-Loop Subspace Identification
Consider a time-discrete linear time-invariant dynamic system described by the state space models in the innovation form:
kkkk
kkkk
eDuCxy
KeBuAxx
++=++=+1 (1)
where mk Ru ∈ and l
k Ry ∈ denote the input and output signals, respectively, and nk Rx ∈ is the
vector of states. lk Re ∈ is zero-mean Gaussian white noise and independent of past input and output
data. A, B, C, D and K are matrices with appropriate dimensions.
1.2 Open-Loop Subspace Identification Problem
The subspace identification problem is: given [ ]nduuu ,..,1= and [ ]ndyyy ,..,1= a set of input-output
measurements, determine the order n of the unknown system, the system matrices (A, B, C, D) up to within a similarity transformation and Kalman filter gain K [15].
1.3 Subspace Matrix Equation
Making successive iterations in equation (1), one can derive the following matrix equation:
fsif
difif EHUHXY ++Γ= (2)
where subscript f stands for the “future” and p for the “past”. For the definition of the matrices diH and s
iH given in (2), see [15]. The past and future input block-Hankel matrices are defined as:
=
−+−
−
21
10
jii
j
p
uu
uu
U
K
MKM
K
(3)
where mixNfp RUU ∈, . The output and noise innovation block-Hankel matrices lixN
fp RYY ∈, and mixN
fp REE ∈, , respectively, are defined in a similar way in (3).
The states are defined as:
[ ]100 ,.., −== jp xxXX (4)
[ ]1,.., −+== jiiif xxXX (5)
The extended observability matrix iΓ is given by:
=Γ
−1i
i
CA
CA
C
K
(6)
The orthogonal projection of the row space ofxA into the row space of
xB is:
xTxx
Txxxx BBBBABA *)(/ = (7)
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where *)(• denotes the Moore-Penrose pseudo-inverse of the matrix )(• .
The projection of the row space ofxA into the orthogonal complement of the row space of
xB is:
xxxxx BAABA // −=⊥ (8)
The oblique projection of the row space of G along the row space H into the row space of J is:
JHJHGJG H ⋅⋅= ⊥⊥ *)/()/(/ (9)
Properties of the orthogonal and oblique projections:
0/ =⊥xx AA (10)
0/ =xAx CAx
(11)
For a proof, see [15].
2. PROPOSED IDENTIFICATION METHOD
2.1 MON4SID Identification Method
In this subsection, it is presented the MON4SID method. To solve the problem in section 1.2, it is used the POMOESP method to calculate the extended observability matrix
iΓ and the N4SID method
is employed to calculate the matrices A, B, C, D through the least squares method. Therefore, it is necessary to eliminate the last two terms in the right side of equation (2). That is done in two steps: first, eliminating the term
fdi UH in (2), performing an orthogonal projection of equation (2) into the
row space of ⊥fU , which yields:
⊥⊥⊥⊥ ++Γ= ffsiff
diffiff UEHUUHUXUY //// (12)
And by the property (10), equation (12) can be simplified to:
⊥⊥⊥ +Γ= ffsiffiff UEHUXUY /// (13)
Second, to eliminate the noises in (13), it is defined an instrumental variable TTp
Tp YUZ )(= .
Multiplication of (13) by Z yields:
ZUEHZUXZUY ffsiffiff
⊥⊥⊥ +Γ= /// (14)
As it is assumed that the noise is uncorrelated with input and output past data [17], which means that 0/ =⊥ ZUE ff
. Therefore, (14) is written as:
fiff XZUY ˆ/ Γ=⊥ (15)
In equation (15), fff XZUX ˆ/ =⊥ is the estimate of the Kalman filter state. Equation (15) indicates
that the column space of iΓ can be calculated by the SVD decomposition of ZUY ff
⊥/ . For further
details see [17]. iΓ , given in (15), can be derived from a simple LQ factorization of a matrix
constructed from the block-Hankel matrices fU ,
pU and fY ,
pY , in the form:
=
3
2
1
333231
2221
11
0
00
Q
Q
Q
LLL
LL
L
Y
Z
U
f
p
f (16)
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and the orthogonal projection in the left side of (15) can be computed by matrix 32L [17]. The SVD of
32L can be given as:
T
T
Tn USV
V
V
S
SUUL =
=
2
1
22132 0
0)( (17)
The order n of the system is equal to the number of non-zero singular values in S . The column space of
1U approximates that of iΓ in a consistent way [17], that is:
1Ui =Γ (18)
The system (1) can be written as:
+
=
+
2
1
~~
r
r
U
X
DC
BA
Y
X i
ii
i
ii
i (19)
In equation (19), suppose (ideally) that 1
~+iX and
iX~ are given, then the matrices system (A, B, C, D)
could be computed through the least squares method. Therefore, the problem now is to find the state sequences.
pUfi ZYf
/=Θ is the oblique projection [15], which is achieved by performing an oblique projection
of equation (2) along the row space fU onto the row space of
pZ , that is:
pUfsipUf
dipUfipUf ZEHZUHZXZY
ffff//// ++Γ= (20)
It is easy to see that the last two terms of equation (20) are zero, by the property of the oblique projection, equation (11); and by the assumption that the noise is uncorrelated with input and output past data [15]. Thus, equation (20) can be simplified to:
iipUf XZYf
~/ Γ= (21)
where pUfi ZXX
f/
~ = . Then, equation (21) is written as:
iii X~Γ=Θ (22)
The oblique projection iΘ given in equation (22) can be computed from (16) by:
==Θ −
2
12221
12232 )()(/
Q
QLLLLWUY pffi
. (23)
An estimate of the state sequence X is given by:
pi ZLLX 12232
* )()( −Γ= . (24)
The following matrices are defined: )1:1(:,~ −= NXX i
, ):2(:,~
1 NXX i =+ . Thus, the matrices system
can be estimated from equation (19). To estimate K see [15] or [17].
2.2 Closed-loop Identification Method
Figure 1 shows a typical standard closed-loop system, where P and C denote respectively the plant and the controller,
kr is the exogenous input, ku the input control,
ky the plant output, kw the plant
disturbance and kv the plant noise.
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Figure 1. Closed-loop system.
P is given by equation (1) and C can be described by the following state equation:
)(
)(1
kkckck
kkckck
yrDxCu
yrBsAs
−+=−+=+ (25)
where cA ,
cB , cC and
cD are matrices with appropriate dimensions.
2.3 Closed-loop Subspace Identification Problem
Given ),,( kkk yur , a set of input/output measurements finite data, of a well-posed problem [7], one
considers the problem of identifying the deterministic part of the plant, that is, one determines the order n of the unknown system, the matrices system (A, B, C, D) up to within a similarity transformation.
2.4 Identification by Joint Input/Output Approach
The objective of this paper is to obtain a state space model of the deterministic part of the plant P, based on finite measurement data ),,( kkk yur . The present problem is practically the same as it was
exposed in [17], but the approach is quite different, as it is not necessary to know any information about the controller.
Using equations (1) and (25), it is possible to obtain a global state space model [17]:
kkkk
kkkk
rDC
rBA
Λ++Ψ=Φ
Ξ++Ψ=Ψ +~~
~~1 (26)
where TTk
Tkk sx ][=Ψ , TT
kTkk yu ][=Φ .
kk ΛΞ , are noises and DCBA~
,~
,~
,~ are matrices with appropriate
dimensions.
The method MON4SID is applied to find an estimate of the matrices DCBA~
,~
,~
,~ , and based on them, to
estimate kΦ . Once
kΦ is known, it is easy to compute the matrices of the plant. To do so, the method
POMOESP [18] is used.
3. SIMULATION
In this section, we provide a simulation example to evaluate the performance of the MON4SID algorithm and compare it with other existing identification algorithms: PEM, N4SIDC and ARXS. N4SIDC here denotes the algorithms of Van Overschee and De Moor (1997) and ARXS the algorithm of [11]. This example was used by [4], [7], [16], [17].
kw kv
P C + -
kr ku ky
+
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The plant is a discrete time model of a laboratory plant setup of two circular plates rotated by an electrical servo motor with flexible shafts. For further details, see [3]. The model of the plant is given by equation (1), where:
−
−=
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
86.0
4
83.7
09.8
4.4
A
−
= −
02.0
3.3
59.18
99.12
98.0
10, 3B
=
0
0
0
0
1
, TC,
−
−=
86336.0
0146.4
515.7
64.6
3.2
K and 0=D
ke is the white noise, which generates the disturbance on the plant, with standard deviation equal to 0
(for the case of deterministic system) and 0.01 (to denote a system of high noise). The controller has a state space description as in the equation (25), where:
−−
=
0
1
0
39.0
1
0
0
75.1
0
1
0
11.3
0
0
1
65.2
cA
=
0
0
0
1
, cB ,
−
−
=
2521.0
7625.0
8629.0
4135.0
TCC
and 61.0=cD
PRBS was used as an exogenous input signal, that is, persistently exciting of any finite order. There have been collected 3000 samples and the number of block rows 20=i .
The simulation results in a closed-loop deterministic identification without noise, shown in figure 2, where the order of the plant is n = 5. From the Figure 2, one can observe that all algorithms had a good performance, except the algorithm N4SIDC.
Figure 2. Bode plots of the plant P, to n=5 and no noise.
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To see an advantage of the proposed algorithm, a white noise is added to the plant, with measurement noise variance 0.01. The comparison results are shown in Figure 3
Figure 3. Bode plots of the plant P, to n=7.
From Figure 3, one can see that the algorithm MOSID performs better in the presence of noise. The order for identification of the plant is n=7. Based on Figure 3, one can say that direct identification models do not have a good performance for closed-loop identification in the presence of noises.
6. CONCLUSIONS
In this work, it was presented the MON4SID algorithm, which uses LQ factorization in the same way as in the MOESP method, which is used to compute the oblique and orthogonal projections; these projections are used to compute the state sequence and the extended observability matrix, respectively. The past and future state sequences are computed from the state sequences, which have only one initial state. It does not happen in the N4SID method, because for each oblique projection (
iΘ and 1+Θ i) different state sequences (
1
~X and
1
~+iX ) are computed, generating a problem of bias in
the estimates.
This algorithm was compared with three identification algorithms (PEM, N4SIDC, ARXS) when applied to a simulated example, which was used in [16] to identify a plant model in discrete time state space. Their results were compared by means of Bode plot. The algorithm MON4SID presented good performance in all the cases. This algorithm has an advantage over the N4SIDC algorithm once does not need any knowledge about the controller.
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